Discrete Weibull distribution
In the original paper by Nakagawa and Osaki they used the parametrization making the cmf with . Setting makes the relationship with the geometric distribution apparent.
The continuous Weibull distribution has a close relationship with the Gumbel distribution which is easy to see when log-transforming the variable. A similar transformation can be made on the discrete-weibull.
Define where (unconventionally) and define parameters and . By replacing in the cmf:
We see that we get a location-scale parametrization:
which in estimation-settings makes a lot of sense. This opens up the possibility of regression with frameworks developed for weibull-regression and extreme-value-theory. 
- Nakagawa, Toshio; Osaki, Shunji (1975). "The discrete Weibull distribution". IEEE Transactions on Reliability. 24: 300–301.
- Scholz, Fritz (1996). "Maximum Likelihood Estimation for Type I Censored Weibull Data Including Covariates". ISSTECH-96-022, Boeing Information & Support Services. Retrieved 26 April 2016.
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